Question

let X be a binomial random variable with n=8 and p=0.6. find the following

(a) the mean of X

(b) the standard deviation of X

(c) P(X<4)

(d) P(X>5)

(e) P(3<X<6)

Answer 1

X ~ Binomial (n,p)

Where n = 8 , p = 0.6

Binomial probability distribution is

P(X) = ⁿC_x p^x (1-p)^n-x

a)

Mean = n * p

= 8 * 0.6

= 4.8

b)

Standard deviation = Sqrt( np(1-p) )

= sqrt( 8 * 0.6 * 0.4)

= 1.38564

c)

P( X < 4) = P( X <= 3)

= P( X = 0) +P( X = 1) +P( X = 2) +P( X = 3)

= ⁸C₀ 0.6⁰ 0.4⁸ +⁸C₁ 0.6¹ 0.4⁷ +⁸C₂ 0.6² 0.4⁶ +⁸C₃ 0.6³ 0.4⁵

= 0.1737

d)

P( X > 5) = P( X >= 6)

= P( X = 6) + P( X = 7) + P( X = 8)

= ⁸C₆ 0.6⁶ 0.4² +⁸C₇ 0.6⁷ 0.4 +⁸C₈ 0.6⁸ 0.4⁰

= 0.3154

e)

P( 3 < X < 6) = P( X = 4) + P( X = 5)

= ⁸C₄ 0.6⁴ 0.4⁴ +⁸C₅ 0.6⁵ 0.4³

= 0.5109